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\footnotesize
$\MathRightMid{r}{$
{\tgbf Copyright \textcopyright\ 2024 Songbingzhi628}$\\$
{\tgbf Email: 13012057210@163.com}$\\[-10pt]\\$
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$\hText{$
This work is licensed under the terms of the CC BY-NC-SA 4.0 International License$\\$ (\url{https://creativecommons.org/licenses/by-nc-sa/4.0}). This license requires that$\\$
reusers give credit to the creator. It allows reusers to distribute, remix, adapt, and$\\$
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identical terms. {\tgsl All images except for `by-nc-sa.png' in this manual are licensed under CC0.}$}$\par\vspace{10pt}
\SepLine[4pt]

%\centerline{\Large 简介}\vspace{6pt}\par
{\footnotesize 这是我个人用于复习的「 {\tgsc Linear Algebra Done Right 3E/4E, by Sheldon Axler} 」笔记。一本习题选答与课文补注。\par
习题我有几不摘：（1）太过小儿科的不摘，比如1.A节；（2）虽然有难度，但习题都是一遍过，或者过不去明显是因为和教材无关的知识门槛；（3）对书中定理显而易见的应用题；（4）算术题；（5）有前章基础者可轻易在书中原地解决的；（6）暂时没想到一题多解的。\par\vspace{4pt}
{\footnotesize 习题解答中，有我完全独立思考得出的，有抄 \url{https://linearalgebras.com/} （这个网站的稳定性非常差，我每每以为它被嘎了）, 有抄 \href{https://math.stackexchange.com/}{stackexchange}, 有抄 \textcolor{blue}{LADR2eSolutions（By Axler）.pdf } ，有抄最新的 \textcolor{blue}{LADR4eSolutions经典最全（By Axler？）.pdf}（ {\scriptsize 这些文档的许可证件，除最后这个找不到/没有指明外，都允许复制/引用）}，还有请教别人，乃至请教AI得出来的。不过别担心，我请教AI时，要么是要找出一个命题的反例（典型的就是有限维成立而无限维不成立的反例），要么是在已经得到正确解答后想看看有没有别的思路。\par\vspace{4pt}
课文补注就是一些\textbf{Tips}和\textbf{Notes}，有警示、\!简化方法、\!对正文的直接补充、\!还有从习题中总结出的二级结论。我将课文中简单而便利的定义（比如3.E节的积空间）前置（比如1.C节）使用。这相当于更改了原书的内容顺序。因为使用中文会给我编撰这份笔记带来额外的中英文输入法切换的工作成本，况且对于专业学习者，直接使用英文不会造成任何困扰。但英文词句的冗长性拖慢我编撰/复习的效率，所以我对做了简写表。我很马虎的，所以会有很多小错误，如有发现就快告诉我！我自己也会不定期抓错玩。\par\vspace{4pt}
题目标为正常数字{\tgbfxx N}的，为3e某章某节第{\tgbfxx N}题（有个别题是4e又删去或增添设问的，仍用3e题号）。因为要面向以第三版为纸面教材的学习者，所以为了避免混淆，摘录4e题目时将题号略去或小字号括弧内标注；而一些新增过多内容的章节会全盘使用4e题号。题目顺序会有调换。除了原书4e新加入的章节外，均使用原书3e的索引。这也许对4e的使用者很不友好。然而4e并不算是\textbf{全盘}将3e覆盖后又增加新内容。比如3e第9章和第10章B节，复化的思想在4e中删减很多。}\par\vspace{-18pt}
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	\tgbf{Email:}&\tgbf{13012057210@163.com}\\
	\tgbf{Bilibili:}&\tgbf{H-U\_O}\\
	\tgbf{Gitee/GitHub:}&\tgbf{Songbingzhi628}\\
\end{tabular}
\end{flushright}
}\par\vspace{-24pt}
\centerline{\Large 作者序}\vspace{6pt}\par
{\small 我目前还没有能力和资格评论原书好坏以及线性代数课程教材选用的问题。但作为原书的学习者，我可以说：\par\vspace{2pt}
相较于（其他课程的）其他教材，以LADR作为\textbf{自学读本}的\textbf{精学}计划，往往在执行中出现一次又一次的时间误判/超时，{\footnotesize 比如我最开始计划$40\times 8$h完成LADR的精学，差不多是一天（$8$h）完成一节，还有额外的复习时间。但在实际学习中，（刨去笔记的工夫）完成到一半时，发现已经耗费了约$35\times 8$h，于是我不得不重新估计LADR精学所需的总时间为$70\times 8$h。出乎意料的是，后一半学习异常高效、\!顺利，也许前一半的学习困难多半是因为基础能力太差。最后实际用时约$60\times 8$h，刨去\TeX 笔记的代码工夫；若算上这个，则约$95\times 8$h，这里仍刨去了初学\TeX 和个人定制宏的时间，\TeX 笔记用时很多还是因为不熟练和强迫症。}\par\vspace{4pt}
这一点对于有学时/学期限制/应试要求的线性代数初学者来说很不安全。更主观地讲，这是因为LADR更像是一本参考手册，而不是一本细致入微的自学读本；如果把LADR作为初学线性代数第一教材和自学读本来学习，会面临不小的困难。\par\vspace{4pt}
以上或许能劝退相当一部分打算入门的线性代数初学者。S.Axler说这本书作为第二遍学习线性代数的教材更合适。我认为理由就是，在校的科班生第二遍学习线性代数时，也已经了解过一众经典的数学分支，这些经历会化作一个叫“mathematical maturity”的东西，让他们面对LADR的课文和习题不再少见多怪、茫然无措。据此，我进一步认为，对于完全的初学者，想要完成LADR的精学，要么有很好的天赋，要么拿出足够的耐心和毅力。就我个人来说：课文一次看不懂，就多看几遍，一天看不懂，就分三天看；习题一个小时做不出来，就隔六个小时再尝试，一天做不出来，就隔天再尝试。这确实让我收获了独特的学习体验和能力，我迄今也无法在别处得到，因此我很珍视LADR，我愿意为此编撰一份电子辅助书并Open到网上。{\footnotesize 实际的时间开销包括了很多不相干的额外项目：初学\LaTeX、\!调整代码架构、\!为我的视觉强迫症让步、\!了解许可证选用，诸如此类的各种波折。大把的时间花销还因为：早期的学习态度还不够主动，导致太多'一遍过'的习题被摘录到这里；没有独立编撰大型文档的经验和模板，可能会强迫症似地纠结散乱的格式和对齐。}\par\vspace{4pt}
{\footnotesize 我在学习过程中碰到了很多重大误区：\textbf{第一章中}，我一开始误认为$W=\complement_V U\cup\def\envFont{\footnotesize}\zeroSubs$是唯一使得$W\oplus U=V$的子空间，但这压根就不是子空间，而且C节习题中也提示这样的子空间$W$不唯一。\textbf{第二章中}，我随意地将“线性无关的序列”等同于有/无限维向量空间的基，没有任何理论依据，我也并不懂什么选择公理。\textbf{第三章B到D节中}，我总觉得子空间是超脱有限维的存在；因为放不下第二章无限维向量空间的基的情结，我刻意寻找那些避开涉及基的解法，一些臆测的结论和容易就找到反例。\textbf{第三章E节中}，我似乎对商空间有什么误解，觉得$v+U=v'+U$如同变戏法一样，把$v$中一切带有$U$的部分抹除掉，让$v$变得\texttt{纯粹}独立于$U$，为此我还单门发明了$\text{Pure}\,V\XSlash U$并试着证明一些命题，甚至用它发现了F节23题无限维情况下不依赖基和同构的解法。后来我猛然发现我最开始的想法多么荒诞，却仍然放不下$\text{Pure}\,V\XSlash U$的情结。这些挫折让我思维变得更加缜密，于是在内化抽象的\textbf{第三章F节}时比想象中的要顺利，及时避开了一些误区。作为回报，我仅用了两小时就完结了\textbf{第六章C节}\!（包括4E）\!除计算题以外的所有内容。\par\vspace{4pt}
{\small 最后，我愿说，与“mathematical maturity”相反的“mathematical naivety”，反而是一种财富。前期因为这种naivety，许多荒诞不经的奇思妙想就能在脑海中飞驰；后期因为上升来的maturity，我就再也没有像前期那样灵感乍现过了。}
}\par\vspace{4pt}
}
\pagebreak

%\centerline{\textsc{\large Goto}}\par\vspace{8pt}
%
%\small
%% To see this cleaner, type Ctrl+- a few times.
%\centerline{
%\begin{tabular}{ | c | c c c c c c || c | c c c c c c | }
%\hline
%1 & 				& \Lch{1B}{B}		& \Lch{1C}{C}	&				&				&				& 6 & \Lch{6A}{A}	& \Lch{6B}{B}	& \Lch{6C}{C}	& \Lch{6D}{D}	&		&				\\
%2 & \Lch{2A}{A}		& \Lch{2B}{B}		& \Lch{2C}{C}	&				&				&				& 7 & \Lch{7A}{A}	& \Lch{7B}{B}	& \Lch{7C}{C}	& \Lch{7D}{D}	&		& \Lch{7F}{\;F*}\\
%3 & \Lch{3A}{A}		& \Lch{3B}{B}		& \Lch{3C}{C}	& \Lch{3D}{D}	& \Lch{3E}{E}	& \Lch{3F}{F}	& 8 & \Lch{8A}{A}	& \Lch{8B}{B}	& \Lch{8C}{C}	& \Lch{8D}{D}	&		&				\\
%\Lch{4O}{4} &				&					&				&				&				&				& 9 & \Lch{9A}{A}	& \Lch{9B}{B}	&				&				&		&				\\
%5 & \Lch{5A}{A}		& \Lch{5BI}{\;$\TXT{B}^\TXT{I}$} & \Lch{5BII}{\;\,$\TXT{B}^\TXT{II}$} & \Lch{5C}{C} & \Lch{5E}{\;E*} & & 10 & \Lch{10A}{A} & \Lch{10B}{B} & & & & \\
%\hline
%\end{tabular}
%}\par\vspace{40pt}

\footnotesize
\def\formGap{$\\\;\\$}
\centerline{{\Large A{\small BBREVIATION} T{\small ABLE}}}\vspace{14pt}\par
$\hMath{c}{\left.}{\right.}{$
	{\tgbf\normalsize A B}$\\$
\begin{tabularx}{0.25\textwidth}{
		| r |
		| >{\raggedright\arraybackslash}X | }
	\hline
	abs&			absolute						\\
	add&			addi(tion)(tive)				\\
	adj&			adjoint							\\
	algo&			algorithm						\\
	arb&			arbitrary						\\
	assoc&			associa(tive)(tivity)			\\
	asum&			assum(e)(ption)					\\
	\hline
%\end{tabularx}\formGap
%	{\tgbf\normalsize B}$\\$
%\begin{tabularx}{0.2\textwidth}{
%		| r |
%		| >{\raggedright\arraybackslash}X | }
%	\hline
	becs&			because							\\
	bss&			basis							\\
	bses&			bases							\\
	$B_V$&			basis of $V$					\\
	\hline
\end{tabularx}\formGap
	{\tgbf\normalsize E}$\\$
\begin{tabularx}{0.28\textwidth}{
		| r |
		| >{\raggedright\arraybackslash}X | }
	\hline
	-ec&			-ec(t)(tor)(tion)(tive)		\\
	eig-&			eigen-						\\
	elem&			element(s)					\\
	ent&			entr(y)(ies)				\\
	equa&			equality					\\
	equiv&			equivalen(t)(ce)			\\
	exa&			example						\\
	exe&			exercise					\\
	exis&			exist(s)(ing)				\\
	existns&		existence					\\
	expo&			exponent					\\
	expr&			expression					\\
	\hline
\end{tabularx}\formGap
	{\tgbf\normalsize L}$\\$
\begin{tabularx}{0.2\textwidth}{
		| r |
		| >{\raggedright\arraybackslash}X | }
	\hline
	liney&			linear(ly)				\\
	linity&			linearity				\\
	len&			length					\\
	low-&			lower-					\\
	\hline
\end{tabularx}\formGap
	{\tgbf\normalsize R}$\\$
\begin{tabularx}{0.3\textwidth}{
		| r |
		| >{\raggedright\arraybackslash}X | }
	\hline
	recurly&		recursively				\\
	repeti&			repetition(s)			\\
	representa&			representation(s)	\\
	req&			require(s)(d)/requiring	\\
	respectly&		respectively			\\
	restr&			restrict(ion)(ive)(ing)	\\
	rev&			revers(e(s))(ed)(ing)	\\
	rotat&			rotation				\\
	\hline
\end{tabularx}\formGap
$}\hMath{c}{\left.}{\right.}{$
	{\tgbf\normalsize C}$\\$
\begin{tabularx}{0.34\textwidth}{
		| r |
		| >{\raggedright\arraybackslash}X | }
	\hline
	ch&				characteristic					\\
	closd&			closed under					\\
	coeff&			coefficient						\\
	col&			column							\\
	combina&		combination						\\
	commu&			commut(es)(ing)(ativity)		\\
	cond&			condition						\\
	conjug&			conjugat(e)(ing)(ion)			\\
	corres&			correspond(s)(ing)				\\
	const&			constant						\\
	conveni&		convenience						\\
	convly&			conversely						\\
	countexa&		counterexample					\\
	countclockws&	counterclockwise				\\
	ctradic&		contradict(s)(ion)				\\
	ctrapos&		constrapositive					\\
	\hline
\end{tabularx}\formGap
	{\tgbf\normalsize F G H}$\\$
\begin{tabularx}{0.26\textwidth}{
		| r |
		| >{\raggedright\arraybackslash}X | }
	\hline
	factoriz&		factorizaion				\\
	fini&			finite(ly)					\\
	finide&			finite-dimensional			\\
	\hline
%\end{tabularx}\formGap
%	{\tgbf\normalsize G}$\\$
%\begin{tabularx}{0.22\textwidth}{
%		| r |
%		| >{\raggedright\arraybackslash}X | }
%	\hline
	g-eig-&				generalized eig-		\\
	G disk&				Gershgorin disk			\\
	\hline
%\end{tabularx}\formGap
%	{\tgbf\normalsize H}$\\$
%\begin{tabularx}{0.2\textwidth}{
%		| r |
%		| >{\raggedright\arraybackslash}X | }
%	\hline
	homo&			homogeneity					\\
	hypo&			hypothesis					\\
	\hline
\end{tabularx}\formGap
%	{\tgbf\normalsize J}$\\$
%\begin{tabularx}{0.2\textwidth}{
%		| r |
%		| >{\raggedright\arraybackslash}X | }
%	\hline
%\end{tabularx}\formGap
%{\tgbf\normalsize K}$\\$
%\begin{tabularx}{0.2\textwidth}{
%		| r |
%		| >{\raggedright\arraybackslash}X | }
%	\hline
%\end{tabularx}\formGap
	{\tgbf\normalsize M N}$\\$
\begin{tabularx}{0.30\textwidth}{
		| r |
		| >{\raggedright\arraybackslash}X | }
	\hline
	max&			maxi(mal(ity))(mum)		\\
	min&			mini(mal(ity))(mum)		\\
	multi&			multipl(e)(icati-on/ve)	\\
	multy&			multiplicity			\\
	\hline
%\end{tabularx}\formGap
%	{\tgbf\normalsize N}$\\$
%\begin{tabularx}{0.2\textwidth}{
%		| r |
%		| >{\raggedright\arraybackslash}X | }
%	\hline
	nilp&			nilpotent				\\
	non0&			nonzero					\\
	nonC&			nonconst				\\
	notat&			notation(al)			\\
	\hline
\end{tabularx}\formGap
	{\tgbf\normalsize S}$\\$
\begin{tabularx}{0.24\textwidth}{
		| r |
		| >{\raggedright\arraybackslash}X | }
	\hline
	seq&			sequence				\\
	simlr&			similar(ly)				\\
	singval&		singular value			\\
	solus&			solution				\\
	sp&				space					\\
	stmt&			statement				\\
	std&			standard				\\
	supp&			suppose					\\
	surj&			surjectiv(e)(ity)		\\
	suth&			such that				\\
	symm&			symmetry				\\
	\hline
\end{tabularx}\formGap
$}\hMath{c}{\left.}{\right.}{$
	{\tgbf\normalsize D}$\\$
\begin{tabularx}{0.31\textwidth}{
		| r |
		| >{\raggedright\arraybackslash}X | }
	\hline
	def&			definition						\\
	deg&			degree							\\
	dep&			dependen(t)(ce)					\\
	deri&			derivative(s)					\\
	diag&			diagonal(iza-ble/ility/tion)	\\
	diff&			differentia(l)(ting)(tion)		\\
	dim&			dimension(al)					\\
	disti&			distinct						\\
	distr&			distributive propert(ies)(ty)	\\
	div&			div(ide)(ision)					\\
	\hline
\end{tabularx}\formGap
{\tgbf\normalsize I}$\\$
\begin{tabularx}{0.31\textwidth}{
		| r |
		| >{\raggedright\arraybackslash}X | }
	\hline
	id&				identity					\\
	immed&			immediately					\\
	induc&			induct(ion)(ive)			\\
	infily&			infinitely					\\
	inje&			injectiv(e)(ity)			\\
	inv&			inver(se)(tib-le/ility)		\\
	invar&			invariant					\\
	invard&			invariant under				\\
	invarsp&		invariant subspace			\\
	invarspd&		invariant subspace under	\\
	iso&			isomorph(ism)(ic)			\\
	isomet&			isometry					\\
	\hline
\end{tabularx}\formGap
	{\tgbf\normalsize O P Q}$\\$
\begin{tabularx}{0.2\textwidth}{
		| r |
		| >{\raggedright\arraybackslash}X | }
	\hline
	optor&			operator				\\
	othws&			otherwise				\\
	orthog&			orthogonal				\\
	orthon&			orthonormal				\\
	\hline
%\end{tabularx}\formGap
%	{\tgbf\normalsize P}$\\$
%\begin{tabularx}{0.2\textwidth}{
%		| r |
%		| >{\raggedright\arraybackslash}X | }
%	\hline
	poly&			polynomial				\\
	posi&			positive				\\
	prod&			product					\\
	\hline
%\end{tabularx}\formGap
%	{\tgbf\normalsize Q}$\\$
%\begin{tabularx}{0.2\textwidth}{
%		| r |
%		| >{\raggedright\arraybackslash}X | }
%	\hline
	quad&			quadratic				\\
	quotient&		quot					\\
	\hline
\end{tabularx}\formGap
	{\tgbf\normalsize T U V W X Y Z}$\\$
\begin{tabularx}{0.22\textwidth}{
		| r |
		| >{\raggedright\arraybackslash}X | }
	\hline
	trig&			triangular				\\
	trslate&		translate				\\
	trspose&		transpose				\\
	\hline
%\end{tabularx}\formGap
%	{\tgbf\normalsize U}$\\$
%\begin{tabularx}{0.2\textwidth}{
%		| r |
%		| >{\raggedright\arraybackslash}X | }
%	\hline
	uniq&			unique					\\
	uniqnes&		uniqueness				\\
	unit&			unitary					\\
	up-&			upper-					\\
	\hline
%\end{tabularx}\formGap
%	{\tgbf\normalsize V}$\\$
%\begin{tabularx}{0.2\textwidth}{
%		| r |
%		| >{\raggedright\arraybackslash}X | }
%	\hline
	val&			value					\\
	\hline
%\end{tabularx}\formGap
%	{\tgbf\normalsize W}$\\$
%\begin{tabularx}{0.2\textwidth}{
%		| r |
%		| >{\raggedright\arraybackslash}X | }
%	\hline
	-wd&			-ward					\\
	-ws&			-wise					\\
	wrto&			with respect to			\\
	\hline
\end{tabularx}\formGap
%	{\tgbf\normalsize X}$\\$
%\begin{tabularx}{0.2\textwidth}{
%		| r |
%		| >{\raggedright\arraybackslash}X | }
%	\hline
%\end{tabularx}\formGap
%	{\tgbf\normalsize Y}$\\$
%\begin{tabularx}{0.2\textwidth}{
%		| r |
%		| >{\raggedright\arraybackslash}X | }
%	\hline
%\end{tabularx}\formGap
%	{\tgbf\normalsize Z}$\\$
%\begin{tabularx}{0.2\textwidth}{
%		| r |
%		| >{\raggedright\arraybackslash}X | }
%	\hline
%\end{tabularx}\formGap
$}$
\pagebreak

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\newcommand{\BmarkChapter}[2]{\bookmark[level=0,dest=Ch#1#2]{#1.#2}}
\newcommand{\BmarkChapterX}[2]{\bookmark[level=0,dest=Ch#1]{#2}}

\newcommand{\BmarkGeneral}[3]{\bookmark[level=1,dest=#1#2#3]{#3}}
\newcommand{\BmarkGeneralFourthEd}[3]{\bookmark[level=1,dest=#1#24e#3]{4E #3}}

\newcommand{\BM}[1]{\BmarkGeneral{\thisChapter}{\thisSection}{#1}}
\newcommand{\BMFE}[1]{\BmarkGeneralFourthEd{\thisChapter}{\thisSection}{#1}}
\newcommand{\BMFEN}[2][]{\bookmark[level=2,dest=\thisChapter\thisSection4e#2]{#1#2}}

\newcommandx{\BMXStart}[2][2=\thisChapter\thisSection]{\bookmark[level=1,dest=Ch#2]{#1}}
\newcommand{\BMX}[3][2]{\bookmark[level=#1,dest=\thisChapter\thisSection#2]{#3}}

\newcommand{\Anchor}[1]{\hypertarget{#1}{}}

\newcommandx{\BmarkI}[4][3=\thisChapter.\thisSection]{
	\def\thisChapter{#1}
	\def\thisSection{#2}
	\bookmark[level=0,dest=Ch#1#2]{#3}
	#4
}

\BmarkI{1}{B}{
	\BMFE{7}
	\BMX[1]{N1}{Note For Fields}
}

\BmarkI{1}{C}{
	\BM{12}\BM{13}\BM{15}\BM{16}\BM{17}\BM{18}\BM{24}
	\BM{'1}\BM{'2}
	\BMXStart{Tips}\BMX{T1}{1}\BMX{T2}{2}\BMX{T3}{3}
	\BMXStart{Notes}\BMX{NE5}{For Exe (5)}\BMX{NE6}{For Exe (6)}\BMX{N1.45}{For [1.45]}\BMX{NC}{About Complements}
}

\BmarkI{2}{A}{
	\BM{1}\BM{2}\BM{10}\BM{11}\BM{14}\BM{16}\BM{17}
	\BMFE{3}\BMFE{14}
	\BMX[1]{N2.11}{Note For [2.11]}
}

\BmarkI{2}{B}{
	\BM{1}\BM{8}
	\BMFE{9}\BMFE{11}
	\BM{Tips}
	\BMX[1]{N2.34}{Note For Liney Indep Seq and [2.34]}
}

\BmarkI{2}{C}{
	\BM{7}\BM{9}\BM{10}\BM{14}\BM{16}\BM{17}
	\BMFE{10}\BMFE{14}\BMFE{15}\BMFE{16}
	\BM{'1}
	\BMX[1]{T10}{Compare with Exe (10)}
	\BMXStart{Notes}\BMX{N10}{For Exe (10)}\BMX{N15}{For Exe (15)}
}

\BmarkI{3}{A}{
	\BM{11}\BM{13}
	\BMFE{10}\BMFE{11}\BMFE{17}
	\BM{'1}
	\BMXStart{Tips}\BMX{T1}{1}\BMX{T2}{2}
	\BMXStart{Notes}\BMX{NR}{For Restr}\BMX{NFS}{For \ensuremath{\Fbb^S}}\BMX{NLC}{For Complex}
}

\BmarkI{3}{B}{
	\BM{12}\BM{20}\BM{21}\BM{22}\BM{23}\BM{24}\BM{25}\BM{26}\BM{28}\BM{30}
	\BM{'1}
	\BMFE{21}\BMFE{24}\BMFE{27}\BMFE{31}\BMFE{32}
	\BMXStart{Tips}\BMX{T1}{1}\BMX{T2}{2}\BMX{T3}{3}\BMX{T4}{4}\BMX{T5}{5}
}

\BmarkI{3}{C}{
	\BM{5}\BM{6}\BM{9}\BM{10}\BM{11}
	\BMFE{16}\BMFE{17}
	\BMXStart{Tips}\BMX{T1}{1}\BMX{T2}{2}
	\BMXStart{Notes}\BMX{N3.3032}{For [3.30, 32]}\BMX{NTrspose}{For Trspose}\BMX{N3.47}{For [3.47]}\BMX{N3.49}{For [3.49]}\BMX{N4e3.51}{For [4E 3.51]}\BMX{NCRFact}{CR Factoriz}\BMX{NCrankEqRrank}{Rank Col = Row}
}

\BmarkI{3}{D}{
	\BM{3}\BM{4}\BM{5}\BM{6}\BM{7}\BM{8}\BM{9}\BM{10}\BM{16}\BM{18}\BM{19}
	\BMFE{10}\BMFE{17}\BMFE{19}\BMFE{23}
	\BM{'1}\BM{'2}\BM{'3}\BM{'4}
	\BMXStart{Tips}\BMX{T1}{1}\BMX{T2}{2}\BMX{T3}{3}
	\BMXStart{Notes}\BMX{NE15}{For Exe (15)}\BMX{NE34e22}{For Exe (3, 4E 22)}\BMX{N3.60}{For [3.60]}\BMX{N3.62}{For [3.62]}\BMX{N3.65}{For [3.65]}
}

\BmarkI{3}{E}{
	\BM{1}\BM{2}\BM{3}\BM{6}\BM{7}\BM{8}\BM{9}\BM{10}\BM{11}\BM{12}\BM{13}\BM{14}\BM{16}\BM{17}\BM{18}\BM{20}
	\BMFE{14}
	\BM{'1}\BM{'2}
	\BMXStart{Tips}\BMX{T1}{1}\BMX{T2}{2}\BMX{T3}{3}\BMX{T4}{4}
	\BMXStart{Notes}\BMX{N3.79}{For [3.79]}\BMX{N3.7983}{For [3.79, 83]}\BMX{N3.85}{For [3.85]}\BMX{N3.88}{For [3.88]}\BMX{N3.889091}{For [3.88, 90, 91]}
}

\BmarkI{3}{F}{
	\BM{4}\BM{6}\BM{7}\BM{8}\BM{13}\BM{18}\BM{19}\BM{21}\BM{22}\BM{23}\BM{24}\BM{25}\BM{26}\BM{31}\BM{32}\BM{33}\BM{34}\BM{35}\BM{36}\BM{37}
	\BMFE{5}\BMFE{6}\BMFE{17}\BMFE{23}\BMFE{24}\BMFE{25}
	\BM{'1}\BM{'2}\BM{'3}\BM{'4}
	\BMXStart{Tips}\BMX{T1}{1}\BMX{T2}{2}\BMX{T3}{3}\BMX{T4}{4}\BMX{T5}{5}
	\BMXStart{Notes}\BMX{N1}{For Exe (1)}\BMX{N18}{For Exe (18)}\BMX{N3.102}{For [3.102]}\BMX{NP}{For Promotion}\BMX{Ndual}{For subsps of dualsp.}\BMX{N3.106}{For [3.106]}\BMX{N3.108}{For [3.108]}\BMX{N3.109b}{For [3.109](b)}
}

\BmarkI{4}{O}[4]{
	\BM{5}\BM{6}\BM{8}
	\BM{'1}
	\BMFE{3}\BMFE{13}
	\BMXStart{Tips}[4O]\BMX{T1}{1}\BMX{T2}{2}
	\BMXStart{Lemmata}[4O]\BMX{L1}{1}
	\BMXStart{Notes}[4O]\BMX{N4.7}{For [4.7]}\BMX{N4.8}{For [4.8]}\BMX{N4.11}{For [4.11]}
}

\BmarkI{5}{A}{
	\BM{10}\BM{13}\BM{15}\BM{19}\BM{20}\BM{23}\BM{24}\BM{27}\BM{28}\BM{29}\BM{33}\BM{34}\BM{35}\BM{36}
	\BMFE{11}\BMFE{16}\BMFE{37}\BMFE{39}\BMFE{40}
	\BM{'1}\BM{'2}
	\BMXStart{Tips}\BMX{T1}{1}\BMX{T2}{2}\BMX{T3}{3}\BMX{T4}{4}\BMX{T5}{5}
	\BMX[1]{NE2.3}{Note For Exe (2, 3)}
}

\BmarkI{5}{BI}[5.B:I]{
	\BM{5}\BM{9}\BM{11}\BM{16}\BM{17}
	\BMFE{7}\BMFE{8}\BMFE{10}\BMFE{11}\BMFE{13}\BMFE{16}\BMFE{17}\BMFE{18}\BMFE{19}\BMFE{21}\BMFE{23}\BMFE{28}\BMFE{29}
	\BMXStart{Tips}\BMX{T1}{1}
	\BMX[1]{N4e5.33}{Note For [4E 5.33]}
}
\BmarkI{5}{BII}[5.B:II]{
	\BMXStart{4E:5.C}\BMFEN{7}\BMFEN{8}\BMFEN{11}\BMFEN{12}\BMFEN{13}\BMFEN{14}
	\BMXStart{Tips}\BMX{T1}{1}\BMX{T2}{2}
}

\BmarkI{5}{C}[5.C(3E) \& 5.D(4E)]{
	\BM{5}
	\BMFE{13}\BMFE{14}\BMFE{16}\BMFE{18}\BMFE{19}
	\BM{'1}
	\BMXStart{Lemmata}[5C]\BMX{L1}{1}
}

\BmarkI{5}{E}[5.E(4E)]{
	\BM{1}\BM{2}\BM{7}\BM{9}
}

\BmarkI{8}{}[8.A,B,C,D]{
	\BMXStart{Notes}\BMX{N8.19}{For [8.19] Or [4E 8.18]}\BMX{N8.55}{For [8.55]}\BMX{NED6}{For Exe (D.6)}
	\BMXStart{Tips}\BMX{BT1}{B.1}\BMX{BT2}{B.2}\BMX{BT3}{B.3}\BMX{BT4}{B.4}

	\newcommand{\BMXI}[2]{\BMX[1]{#1}{#2}}
	\BMXI{A3}{A.3}\BMXI{A5}{A.5}\BMXI{A6}{A.6}\BMXI{A10}{A.10}\BMXI{A17}{A.17}\BMXI{A18}{A.18}
	\BMXI{A'1}{A.'1}
	\BMXI{A4e3}{4E A.3}\BMXI{A4e4}{4E A.4}\BMXI{A4e12}{4E A.12}\BMXI{A4e15}{4E A.15}\BMXI{A4e16}{4E A.16}\BMXI{A4e18}{4E A.18}\BMXI{A4e24}{4E A.24}

	\BMXI{B5}{B.5}\BMXI{B10}{B.10}
	\BMXI{B4e6}{4E B.6}\BMXI{B4e7}{4E B.7}\BMXI{B4e20}{4E B.20}

	\BMXI{C11}{C.11}\BMXI{C15}{C.15}\BMXI{C18}{C.18}\BMXI{C20}{C.20}

	\BMXI{D8}{D.8}
}

\BmarkI{9}{A}{
	\BM{3}\BM{4}\BM{13}\BM{17}
	\BMXStart{Notes}\BMX{N910}{For [9.10]}\BMX{N912}{For [9.12]}\BMX{N917}{For [9.17]}
}

\BmarkI{6}{A}{
	\BM{3}\BM{6}\BM{21}
	\BM{'1}
	\BMFE{23}
	\BMXStart{Tips}\BMX{T1}{1}\BMX{T2}{2}
}

\BmarkI{6}{B}{
	\BM{2}\BM{9}\BM{10}\BM{11}\BM{12}\BM{14}\BM{16}
	\BM{'1}
	\BMFE{9}\BMFE{10}\BMFE{13}\BMFE{19}
	\BMX[1]{T}{Tips}
}

\BmarkI{6}{C}{
	\BM{8}\BM{10}
	\BMXStart{Tips}\BMX{T1}{1}\BMX{T2}{2}\BMX{T3}{3}\BMX{T4}{4}
}

\BmarkI{7}{A}{
	\BM{17}\BMFE{28}
	\BMX[1]{T}{Tips}
}

\BmarkI{7}{B}{
	\BM{13}\BM{14}
	\BMFE{8}\BMFE{12}\BMFE{20}
}

\BmarkI{7}{C}[7.C(4E)]{
	\BM{20}\BM{22}\BM{23}
	\BMX[1]{NSRI}{Note For Square Root of Id}
}

\BmarkI{7}{D}[7.D(4E)]{
	\BM{2}\BM{5}\BM{11}
	\BMX[1]{T}{Tips}
}

\BmarkI{7}{E}[7.E(4E)]{
	\BM{1}\BM{4}\BM{11}\BM{17}
}

\BmarkI{7}{F}[7.F(4E)]{
	\BM{3}\BM{5}\BM{8}\BM{11}\BM{13}\BM{14}\BM{16}\BM{19}\BM{20}\BM{22}\BM{28}\BM{31}
}

\BmarkI{10A}{}[10.A]{
	\BM{17}
	\BMX[1]{4e10}{4E 8.D.10}
}

\BmarkI{4e9}{}[10.B \& \text{[4E]} 9.A,B,C,D]{
	\BMXI{A2}{A.2}\BMXI{A4}{A.4}
	\BMXI{B1}{B.1}
	\BMXI{C2}{C.2}\BMXI{C5}{C.5}\BMXI{C22}{C.22}
%	\BMXI{D5}{D.5}\BMXI{D9}{D.9}\BMXI{D10}{D.10}\BMXI{D11}{D.11}
}

\include{Ch1-2-3-4-5-8-9A}
\include{Ch6-7-10-4e9}

\end{large}
\end{document}